Remark 1.3 The concept of inner \(\alpha \) -thinness of a set at infinity
was introduced in [1]}.
Referring to Sect.  for details, at this point we only note that \(A\subset \mathbb {R}^n\)  is not inner \(\alpha \) -thin at infinity if and only if \(y=0\)  is inner \(\alpha \) -regular for \(A^*\) , the inverse of \(A\)  with respect to \(\lbrace |x|=1\rbrace \) . Such \(A\)  may in particular be thought of as \(\lbrace x_i>q\rbrace \) , where \(q\in \mathbb {R}\)  and \(x_i\)  is the \(i\) -coordinate of \(x\in \mathbb {R}^n\) , or as \(Q^c:=\mathbb {R}^n\setminus Q\) , \(Q\subset \mathbb {R}^n\)  being bounded (see also Example REF ). If \(A\)  is not inner \(\alpha \) -thin at infinity, it is `rather large' at infinity in the sense that then necessarily \(c_\alpha (A)=\infty \) , whereas the converse is in general not true (see Corollary REF , cf. Example REF  for illustration).

